As shown in the figure,a wire and a 10-Ω resistor are used to form a circuit in the shape of a square with dimensions 20 cm by 20 cm.A uniform but non-steady magnetic field is directed into the plane of the circuit.The magnitude of the magnetic field is steadily decreased from 2.70 T to 0.90 T in a time interval of 96 ms.What is the induced current in the circuit,and what is its direction through the resistor?
A)75 mA,from b to a
B)45 mA,from b to a
C)75 mA,from a to b
D)45 mA,from a to b
E)110 mA,from a to b

Question

As shown in the figure,a wire and a 10-Ω resistor are used to form a circuit in the shape of a square with dimensions 20 cm by 20 cm.A uniform but non-steady magnetic field is directed into the plane of the circuit.The magnitude of the magnetic field is steadily decreased from 2.70 T to 0.90 T in a time interval of 96 ms.What is the induced current in the circuit,and what is its direction through the resistor? 
A)75 mA,from b to a 
B)45 mA,from b to a 
C)75 mA,from a to b 
D)45 mA,from a to b 
E)110 mA,from a to b

Quizplus · Accepted Answer

According to Faraday's Law of Electromagnetic Induction, any change in magnetic flux through a circuit will induce an electromotive force (EMF) and, thus, a current in the circuit. In this case, the magnetic field is decreasing, which means the flux through the circuit is also decreasing. Since the circuit is a square with sides of length 20 cm, each side has a length of 0.2 m. Therefore, the area enclosed by the circuit is (0.2 m)^2 = 0.04 m^2. The magnetic flux through this area is given by:
Φ = B*A*cosθ
where B is the magnetic field, A is the area of the circuit, and θ is the angle between the magnetic field and the surface normal (which is zero in this case, since the magnetic field is perpendicular to the plane of the circuit). Thus, at the beginning of the time interval given, when the magnetic field is 2.7 T:
Φ1 = 2.7 T * 0.04 m^2 * cos(0°) = 0.108 Wb
And at the end of the time interval, when the magnetic field is 0.9 T:
Φ2 = 0.9 T * 0.04 m^2 * cos(0°) = 0.036 Wb
The change in flux during this time interval is therefore:
ΔΦ = Φ2 - Φ1 = -0.072 Wb
The negative sign indicates that the flux is decreasing.
The EMF induced in the circuit is given by Faraday's Law:
EMF = -dΦ/dt
where dΦ/dt is the rate of change of flux over time. Taking the time interval to be 96 ms (0.096 s), we get:
EMF = (-0.072 Wb) / (0.096 s) = -0.75 V
Again, the negative sign indicates that the EMF is opposing the change in flux. This EMF will drive a current through the circuit, which will in turn create its own magnetic field that opposes the original magnetic field. This is known as Lenz's Law.
We can use Ohm's Law to find the current through the 10-Ω resistor:
I = EMF / R = (-0.75 V) / (10 Ω) = -0.075 A = -75 mA
The negative sign indicates that the current is flowing clockwise around the circuit, which means it is flowing from b to a through the resistor. Therefore, the correct answer is A: 75 mA, from b to a.

As Shown in the Figure,a Wire and a 10-Ω Resistor